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Geometry and topology

The UCL Geometry and Topology Group has a broad range of interests.

Differential geometry and geometric analysis.

Michael Singer has a number of interests in the differential geometry of special metrics and related geometric analysis. These include the study of Kähler metrics of constant scalar curvature and related questions about Bergman kernels and density functions; the geometry of self-dual and self-dual Einstein metrics in dimension 4; and also the application of analysis on manifolds with corners to geometric problems on singular and non-compact spaces. He is also interested in the geometry and topology of moduli spaces (for example of Euclidean monopoles).

Jason Lotay works on differential geometry, with a particular focus on 7 and 8-dimensional Riemannian manifolds with exceptional holonomy, their calibrated submanifolds, and related geometries. He is also interested in Lagrangian mean curvature flow. He mainly uses methods from geometric analysis and exterior differential systems, which have applications throughout geometry and other areas in mathematics.

Geometric group theory and low-dimensional topology.

Henry Wilton works on low-dimensional manifolds and their fundamental groups, negatively and non-positively curved groups, profinite groups, group theoretic decision problems and the elementary theories of groups.

F. E. A. Johnson works on the topology of manifolds; low dimensional topology; the D(2) problem, more generally problems involving the fundamental group; Lie groups and their discrete subgroups; homological algebra; geometric invariant theory.

Symplectic and contact topology.

Chris Wendl studies the topology of symplectic and contact manifolds and their relationships to each other, e.g. the problem of classifying the symplectic fillings of a given contact manifold, or of determining whether two contact manifolds are related by a symplectic cobordism. To study these questions, he makes frequent use of pseudoholomorphic curves and the algebraic invariants arising from them, such as Gromov-Witten theory and symplectic field theory. He sometimes also studies the analytical properties of holomorphic curves, especially in dimension 4, where intersection theory plays a vital role.

Jonny Evans works on various problems in symplectic topology. These include: the questions of topological and isotopy rigidity for Lagrangian submanifolds; understanding the relationship between moduli spaces of algebraic varieties and the classifying spaces of their symplectomorphism groups; computing pseudoholomorphic curve invariants in interesting new settings; and studying the symplectic geometry of twistor spaces.

We also have a joint seminar with the geometers at KCL: see here for more information.

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